“POSSE” stands for Publish on your Own Site, Syndicate Elsewhere” (or Everywhere). POSSE is less a formal protocol (like see Webmentions) than a feature you may want to build into your site.
The idea is that, after posting to one’s site, one then pushes copies of, or links to, that new post to assorted third-party sites (Twitter, for instance). This allows readers who prefer those sites to remain aware of your activity without leaving their prefered experience. This feature can be extended to collect responses to the POSSE’d content within the third-party sites (likes, replies and so forth) and feed that back to your site: a list of Tweets and/or Toots that have replied to you underneath your original post, for instance.
You can see an example of this in action (in conjunction with see Webmentions) here.
It stems from the Indieweb principle of owning your data rather than trusting it to a third-party site that may go away or simply decide to deny you access to your own content. It also stems from the hard fact that many of our friends & colleagues are still on such third-party sites such as Twitter or Mastodon: this is a way to connect your content thereto.
Implementations are generally bespoke; indie-org supports this
for the site author via brid.gy (see
see Supporting POSSE).